Graph pattern detection: hardness for all induced patterns and faster non-induced cycles

Abstract

We consider the pattern detection problem in graphs: given a constant size pattern graph H and a host graph G, determine whether G contains a subgraph isomorphic to H. We present the following new improved upper and lower bounds:• We prove that if a pattern H contains a k-clique subgraph, then detecting whether an n node host graph contains a not necessarily induced copy of H requires at least the time for detecting whether an n node graph contains a k-clique. The previous result of this nature required that H contains a k-clique which is disjoint from all other k-cliques of H.• We show that if the famous Hadwiger conjecture from graph theory is true, then detecting whether an n node host graph contains a not necessarily induced copy of a pattern with chromatic number t requires at least the time for detecting whether an n node graph contains a t-clique. This implies that: (1) under Hadwiger's conjecture for every k-node pattern H, finding an induced copy of H requires at least the time of √ k-clique detection and size ω(n √ k/4 ) for any constant depth circuit, and (2) unconditionally, detecting an induced copy of a random G(k, p) pattern w.h.p. requires at least the time of Θ(k/ log k)-clique detection, and hence also at least size n Ω(k/ log k) for circuits of constant depth.• We show that for every k, there exists a k-node pattern that contains a k − 1-clique and that can be detected as an induced subgraph in n node graphs in the best known running time for k − 1-Clique detection. Previously such a result was only known for infinitely many k.• Finally, we consider the case when the pattern is a directed cycle on k nodes, and we would like to detect whether a directed m-edge graph G contains a k-Cycle as a not necessarily induced subgraph. We resolve a 14 year old conjecture of [Yuster-Zwick SODA'04] on the complexity of k-Cycle detection by giving a tight analysis of their k-Cycle algorithm. Our analysis improves the best bounds for k-Cycle detection in directed graphs, for all k > 5.

Streaming Deletion Problems Parameterized by Vertex Cover

“…With this in mind, it is logical that knowing a vertex cover of the input graph can help in our approach. Dalirrooyfard et al [18] relate the hardness of Induced Subgraph Isomorphism to the hardness of k-Clique, which holds under the Hadwiger conjecture. Under ETH, this implies that for almost all H of size k, finding an H in a graph G cannot be done in time n o(k/ log k) .…”

Current Algorithms for Detecting Subgraphs of Bounded Treewidth are Probably Optimal

“…In a different direction, Dalirrooyfard et al [35] design various reductions from k-Clique to Subgraph Isomorphism, among other results. They also present results on the detection of induced subgraphs (we focus on non-induced subgraphs).…”

Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-shortest Induced Paths

“…See [1,14] for the first polynomial-time algorithm for finding an independent set of maximum weight in a graph having no hole of length at least five. See [30] for upper and lower bounds on the complexity of detecting an O(1)-vertex induced subgraph.…”